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A104556
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Matrix inverse of triangle A001497 of Bessel polynomials, read by rows; essentially the same as triangle A096713 of modified Hermite polynomials.
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10
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1, -1, 1, 0, -3, 1, 0, 3, -6, 1, 0, 0, 15, -10, 1, 0, 0, -15, 45, -15, 1, 0, 0, 0, -105, 105, -21, 1, 0, 0, 0, 105, -420, 210, -28, 1, 0, 0, 0, 0, 945, -1260, 378, -36, 1, 0, 0, 0, 0, -945, 4725, -3150, 630, -45, 1, 0, 0, 0, 0, 0, -10395, 17325, -6930, 990, -55, 1, 0, 0, 0, 0, 0, 10395, -62370, 51975, -13860, 1485, -66, 1
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OFFSET
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0,5
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COMMENTS
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Also the Bell transform of (-1)^n if n<2 else 0 and the inverse Bell transform of A001147(n) (adding 1,0,0,... as column 0). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016
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LINKS
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FORMULA
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E.g.f. : (1 - t)*exp(x*(t - t^2/2)) = 1 + (-1 + x)*t + (-3*x + x^2)*t^2/2! + ... - Peter Bala, Apr 08 2013
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EXAMPLE
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Rows begin:
1;
-1, 1;
0, -3, 1;
0, 3, -6, 1;
0, 0, 15, -10, 1;
0, 0, -15, 45, -15, 1;
0, 0, 0, -105, 105, -21, 1;
0, 0, 0, 105, -420, 210, -28, 1;
0, 0, 0, 0, 945, -1260, 378, -36, 1;
0, 0, 0, 0, -945, 4725, -3150, 630, -45, 1; ...
The columns being equal in absolute value to the rows of the matrix inverse A001497:
1;
1, 1;
3, 3, 1;
15, 15, 6, 1;
105, 105, 45, 10, 1;
945, 945, 420, 105, 15, 1; ...
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MATHEMATICA
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With[{nmax = 10}, CoefficientList[CoefficientList[Series[(1 - t)*Exp[x*(t - t^2/2)], {t, 0, nmax}, {x, 0, nmax}], t], x]*Range[0, nmax]!] (* G. C. Greubel, Jun 10 2018 *)
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PROG
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(Sage) # uses[bell_matrix from A264428]
# Adds a column 1, 0, 0, 0, ... at the left side of the triangle.
bell_matrix(lambda n: (-1)^n if n<2 else 0, 9) # Peter Luschny, Jan 19 2016
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CROSSREFS
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Row sums are found in A001464 (offset 1).
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KEYWORD
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AUTHOR
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STATUS
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approved
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